The 4700 uF one is on there for the Amazon one because they rate that board at 6A. If you draw current like that from a bridge rectifier without a huge filter cap, you will have a ton of ripple in the output DC. Even with 4700uF, the ripple will be something like 6 volts of difference between the dips and peaks of the output. It all depends on how much ripple you can tolerate. Frankly, as Peteski said, ripple in a supply that's just for LEDs isn't going to matter. You'll never see it visually with your eyes and it won't cause any damage to the device. But for a board that's promising the be a "filter", I can see why they put that big cap on there. I wouldn't worry about the startup inrush current for that big capacitor. 4700 is big, but not that big, really.
If you buy that Amazon one, I'd be curious to know if it really comes with a Nichicon cap on it. That's one of the nicer premium brand electrolytics and usually cheap little boards through Amazon from China do not use components like that.
Yes, we are rectifying a square-wave signal so there is none of that sinewave ripple. There might be some small dropouts when the DCC signal reverses polarity, but those will be really insignificant (for use with LEDs).
